Theorem abianfplem 3961

Description: Lemma for abianfp 3962. We prove by transfinite induction that if F has a fixed point x, then its iterates also equal x. This lemma is used for the "trivial" direction of the main theorem.

Hypotheses

Ref	Expression
abianfp.1	|- A e. V
abianfp.2	|- G = rec({<.z, w>. | w = (F` z)}, x)

Assertion

Ref	Expression
abianfplem	|- (v e. On -> ((F` x) = x -> (G` v) = x))

Distinct variable groups:   x,v,A   x,z,w,F,v   v,G

Proof of Theorem abianfplem

Step	Hyp	Ref	Expression
1		fveq2 3724	. . 3 |- (v = (/) -> (G` v) = (G` (/)))
2	1	eqeq1d 1483	. 2 |- (v = (/) -> ((G` v) = x <-> (G` (/)) = x))
3		fveq2 3724	. . 3 |- (v = y -> (G` v) = (G` y))
4	3	eqeq1d 1483	. 2 |- (v = y -> ((G` v) = x <-> (G` y) = x))
5		fveq2 3724	. . 3 |- (v = suc y -> (G` v) = (G` suc y))
6	5	eqeq1d 1483	. 2 |- (v = suc y -> ((G` v) = x <-> (G` suc y) = x))
7		abianfp.2	. . . . 5 |- G = rec({<.z, w>. | w = (F` z)}, x)
8	7	fveq1i 3725	. . . 4 |- (G` (/)) = (rec({<.z, w>. | w = (F` z)}, x)` (/))
9		visset 1813	. . . . 5 |- x e. V
10	9	rdg0 3941	. . . 4 |- (rec({<.z, w>. | w = (F` z)}, x)` (/)) = x
11	8, 10	eqtr 1495	. . 3 |- (G` (/)) = x
12	11	a1i 8	. 2 |- ((F` x) = x -> (G` (/)) = x)
13		fvex 3732	. . . . 5 |- (F` (G` y)) e. V
14		ax-17 971	. . . . . 6 |- (u e. x -> A.z u e. x)
15		ax-17 971	. . . . . 6 |- (u e. y -> A.z u e. y)
16		ax-17 971	. . . . . . 7 |- (u e. F -> A.z u e. F)
17		hbopab1 2813	. . . . . . . . . 10 |- (u e. {<.z, w>. | w = (F` z)} -> A.z u e. {<.z, w>. | w = (F` z)})
18	17, 14	hbrdg 3936	. . . . . . . . 9 |- (u e. rec({<.z, w>. | w = (F` z)}, x) -> A.z u e. rec({<.z, w>. | w = (F` z)}, x))
19	7, 18	hbxfr 1563	. . . . . . . 8 |- (u e. G -> A.z u e. G)
20	19, 15	hbfv 3729	. . . . . . 7 |- (u e. (G` y) -> A.z u e. (G` y))
21	16, 20	hbfv 3729	. . . . . 6 |- (u e. (F` (G` y)) -> A.z u e. (F` (G` y)))
22		fveq2 3724	. . . . . 6 |- (z = (G` y) -> (F` z) = (F` (G` y)))
23	14, 15, 21, 7, 22	rdgsucopab 3946	. . . . 5 |- ((y e. On /\ (F` (G` y)) e. V) -> (G` suc y) = (F` (G` y)))
24	13, 23	mpan2 696	. . . 4 |- (y e. On -> (G` suc y) = (F` (G` y)))
25		fveq2 3724	. . . . 5 |- ((G` y) = x -> (F` (G` y)) = (F` x))
26		id 59	. . . . 5 |- ((F` x) = x -> (F` x) = x)
27	25, 26	sylan9eqr 1529	. . . 4 |- (((F` x) = x /\ (G` y) = x) -> (F` (G` y)) = x)
28	24, 27	sylan9eq 1527	. . 3 |- ((y e. On /\ ((F` x) = x /\ (G` y) = x)) -> (G` suc y) = x)
29	28	exp32 377	. 2 |- (y e. On -> ((F` x) = x -> ((G` y) = x -> (G` suc y) = x)))
30		visset 1813	. . . . . . . 8 |- v e. V
31		rdglim2a 3950	. . . . . . . 8 |- ((v e. V /\ Lim v) -> (rec({<.z, w>. | w = (F` z)}, x)` v) = U_y e. v (rec({<.z, w>. | w = (F` z)}, x)` y))
32	30, 31	mpan 695	. . . . . . 7 |- (Lim v -> (rec({<.z, w>. | w = (F` z)}, x)` v) = U_y e. v (rec({<.z, w>. | w = (F` z)}, x)` y))
33	7	fveq1i 3725	. . . . . . 7 |- (G` v) = (rec({<.z, w>. | w = (F` z)}, x)` v)
34	7	fveq1i 3725	. . . . . . . . 9 |- (G` y) = (rec({<.z, w>. | w = (F` z)}, x)` y)
35	34	a1i 8	. . . . . . . 8 |- (y e. v -> (G` y) = (rec({<.z, w>. | w = (F` z)}, x)` y))
36	35	iuneq2i 2580	. . . . . . 7 |- U_y e. v (G` y) = U_y e. v (rec({<.z, w>. | w = (F` z)}, x)` y)
37	32, 33, 36	3eqtr4g 1531	. . . . . 6 |- (Lim v -> (G` v) = U_y e. v (G` y))
38	37	adantr 389	. . . . 5 |- ((Lim v /\ A.y e. v (G` y) = x) -> (G` v) = U_y e. v (G` y))
39		iuneq2 2578	. . . . . 6 |- (A.y e. v (G` y) = x -> U_y e. v (G` y) = U_y e. v x)
40		df-lim 2953	. . . . . . . 8 |- (Lim v <-> (Ord v /\ v =/= (/) /\ v = U.v))
41		3simp2 789	. . . . . . . 8 |- ((Ord v /\ v =/= (/) /\ v = U.v) -> v =/= (/))
42	40, 41	sylbi 199	. . . . . . 7 |- (Lim v -> v =/= (/))
43		iunconst 2572	. . . . . . 7 |- (v =/= (/) -> U_y e. v x = x)
44	42, 43	syl 10	. . . . . 6 |- (Lim v -> U_y e. v x = x)
45	39, 44	sylan9eqr 1529	. . . . 5 |- ((Lim v /\ A.y e. v (G` y) = x) -> U_y e. v (G` y) = x)
46	38, 45	eqtrd 1507	. . . 4 |- ((Lim v /\ A.y e. v (G` y) = x) -> (G` v) = x)
47	46	ex 373	. . 3 |- (Lim v -> (A.y e. v (G` y) = x -> (G` v) = x))
48	47	a1d 12	. 2 |- (Lim v -> ((F` x) = x -> (A.y e. v (G` y) = x -> (G` v) = x)))
49	2, 4, 6, 12, 29, 48	tfinds2 3165	1 |- (v e. On -> ((F` x) = x -> (G` v) = x))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  Vcvv 1811  (/)c0 2280  U.cuni 2503  U_ciun 2566  {copab 2666  Ord word 2947  Oncon0 2948  Lim wlim 2949  suc csuc 2950  ` cfv 3182  reccrdg 3931

This theorem is referenced by:  abianfp 3962

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932

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